Problem 11 - College Algebra

Question

Determine whether each statement is true or false. $$ 4 \geq-7 $$

Step-by-Step Solution

Answer

The statement '4 \(\geq\) -7' is TRUE. Because 4 is greater than -7.

Step 1: Understand the Inequality

The inequality '4 \(\geq\) -7' is read as '4 is greater than or equal to -7'. It tells us that the number on the left side of the inequality (4) is either greater than or equal to the number on the right side (-7).

Step 2: Compare the Values

Compare the numeric values on either side of the inequality. The number 4 is greater than -7. We can also say that 4 is -7 units to the right on the number line, which verifies that 4 is greater than -7.

Step 3: Verify the Statement

Since number 4 is greater than -7, 4 \(\geq\) -7 is a correct statement.

Key Concepts

Number Line

The number line is a visual tool in mathematics that helps us understand the order and position of numbers. Imagine the number line as a horizontal line extending infinitely in both directions. Numbers to the right are greater, and numbers to the left are smaller.

To visualize where numbers like 4 and -7 lie, imagine placing them on this line. The center of commonly used number lines is often zero. In our example, 4 would be positioned to the right of zero, while -7 would be found to the left. This spatial distance more clearly shows that 4 is indeed greater than -7.

Using a number line makes it easy to compare different numbers. It is a simple yet powerful way to grasp concepts like order, size, and distance between numbers.

Greater Than or Equal To

When you see the symbol \( \geq \) in math, it reads as "greater than or equal to." It tells us that the number on the left side of the symbol is either larger or the same as the number on the right.

This symbol combines two ideas: one, that a number can be greater, and two, that it can simply be equal. For example, in the expression 4 \( \geq \) -7, you are checking if 4 is more than -7 or if it is equal to -7.

• If 4 is greater, then the statement holds true.
• If 4 is equal, surprisingly it would also be true, but this is not the case here as 4 is not equal to -7.

The symbol \( \geq \) is common in inequalities, helping compare numerical values with ease.

Comparing Values

Comparing values involves determining the relationship between two numbers—like which is larger or how they relate on a number line.

In the inequality 4 \( \geq \) -7, we compare 4 to -7. Since 4 is located to the right of -7 on the number line, it shows that 4 is greater.

• Think of greater values as being farther to the right.
• Smaller values shift to the left.

Such comparisons follow logical thinking and help solve math problems efficiently. Understanding how values are compared, whether through direct evaluation or using tools like number lines, is essential in verifying whether statements are true or false. This systematic approach allows us to confirm solutions like 4 \( \geq \) -7 and find them accurate.